Discerns Werckmeister P and S? extracts HOW 4th root?

Two original topics concerning Werckmeister and more!

FIRST: Did he distinguish between S and P,
Syntonic and Pythagorean Comma?

SECOND: How does Werckmeister calculate a 4th root?

ad 1 syntonic and pythagorean comma:

In his treatise "Musicalische Temperatur", Werckmeister writes
on page 66, XXIV Cap., in my free English Translation:

If the latter number 262144 is being doubled,then we
do not retrieve the 531441 (its origin from C), but we
rather obtain 524288.

The excess
(between 531441/524288 and 1)
is a comma 81/80
and additionally, a small value occurs,
of 32805/32768 (schisma).

Werckmeister therefore indeed discerns
the Pytagorean from the Syntonic comma and even
calculates the difference! (schisma)

ad 2, fourth root:

On page 37, XVII. Cap., in my free English Translation:
If a Comma
(81/80)
is partitioned into 4 parts, we obtain:

324:323:322:321:320.

The ratio of the outer parts, 324/320 is the Comma (81/80).

COMMENT:

In modern notation, all that boils down to:

81 324 323 322 321
--- = ----*----*----*----
80 323 322 321 320

and an approximation to the fourth root is
established herewith as

324
----
323

Extracting likewise the FIFTH root:
Dividing by 5 the Pythagorean Comma P, in its
approximation as SUPERPARTICULAR RATIO,
74/73, yields

74 370 369 368 367 366
--- = ----*----*----* ----
73 369 368 367 366 365

Thus, the fifth root of P is approximated by 370/369.
370/369 is the superparticular ratio of 369.

The simple (just) intervals are the SUP.RATs, of

1 Octave, 2/1,
2 Fifth 3/2,
3 Fourth 4/3
4 Major Third 5/4, etc.

Werckmeister expresses temperaments as SUP.RATs in his
treatise.

Here is the reason, why Bach's Four Duets measure 369 bars:
369=73+149+39+108. In this composition, Bach has
encoded his system
"Werckmeister/Bach/wohltemperirt".

Kellner, H.A.: How Bach quantified his well-tempered tuning within the Four
Duets. English Harpsichord Magazine, Vol. 4, No. 2, 1986(87), page 21-27

Idem: Barocke Akustik und Numerologie in den Vier Duetten: Bachs "Musicalische
Temperatur". In "Bericht �ber den Internationalen Musikwissenschaftlichen
Kongre� Stuttgart 1985", Hg. Dietrich Berke und Dorothea Hanemann, Kassel 1987,
Seite 439-449

Provided a fifth is reduced by the small value
of 370/369, five such tempered fifths will complete,
together with 7 perfect fifths, 7 Octaves.

370/371 is approximately 4,7 cent.

Here lies the reason, why the first keystroke (bass, of course)
in bar 24 (=all tonalities, major and minor of the well
tempered Clavier) is number 369.
Proof:
Follows from 23*16=368, and
368+1=369.